Theorem fmptapd 7038

Description: Append an additional value to a function. (Contributed by Thierry Arnoux, 3-Jan-2017.) (Revised by AV, 10-Aug-2024.)

Hypotheses

Ref	Expression
fmptapd.a	⊢ (𝜑 → 𝐴 ∈ 𝑉)
fmptapd.b	⊢ (𝜑 → 𝐵 ∈ 𝑊)
fmptapd.s	⊢ (𝜑 → (𝑅 ∪ {𝐴}) = 𝑆)
fmptapd.c	⊢ ((𝜑 ∧ 𝑥 = 𝐴) → 𝐶 = 𝐵)

Assertion

Ref	Expression
fmptapd	⊢ (𝜑 → ((𝑥 ∈ 𝑅 ↦ 𝐶) ∪ {⟨𝐴, 𝐵⟩}) = (𝑥 ∈ 𝑆 ↦ 𝐶))

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝑥,𝑅   𝑥,𝑆   𝜑,𝑥

Allowed substitution hints:   𝐶(𝑥)   𝑉(𝑥)   𝑊(𝑥)

Proof of Theorem fmptapd

Step	Hyp	Ref	Expression
1		fmptapd.c	. . . 4 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → 𝐶 = 𝐵)
2		fmptapd.a	. . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
3		fmptapd.b	. . . 4 ⊢ (𝜑 → 𝐵 ∈ 𝑊)
4	1, 2, 3	fmptsnd 7036	. . 3 ⊢ (𝜑 → {⟨𝐴, 𝐵⟩} = (𝑥 ∈ {𝐴} ↦ 𝐶))
5	4	uneq2d 4102	. 2 ⊢ (𝜑 → ((𝑥 ∈ 𝑅 ↦ 𝐶) ∪ {⟨𝐴, 𝐵⟩}) = ((𝑥 ∈ 𝑅 ↦ 𝐶) ∪ (𝑥 ∈ {𝐴} ↦ 𝐶)))
6		mptun 6576	. . 3 ⊢ (𝑥 ∈ (𝑅 ∪ {𝐴}) ↦ 𝐶) = ((𝑥 ∈ 𝑅 ↦ 𝐶) ∪ (𝑥 ∈ {𝐴} ↦ 𝐶))
7	6	a1i 11	. 2 ⊢ (𝜑 → (𝑥 ∈ (𝑅 ∪ {𝐴}) ↦ 𝐶) = ((𝑥 ∈ 𝑅 ↦ 𝐶) ∪ (𝑥 ∈ {𝐴} ↦ 𝐶)))
8		fmptapd.s	. . 3 ⊢ (𝜑 → (𝑅 ∪ {𝐴}) = 𝑆)
9	8	mpteq1d 5174	. 2 ⊢ (𝜑 → (𝑥 ∈ (𝑅 ∪ {𝐴}) ↦ 𝐶) = (𝑥 ∈ 𝑆 ↦ 𝐶))
10	5, 7, 9	3eqtr2d 2786	1 ⊢ (𝜑 → ((𝑥 ∈ 𝑅 ↦ 𝐶) ∪ {⟨𝐴, 𝐵⟩}) = (𝑥 ∈ 𝑆 ↦ 𝐶))

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1542   ∈ wcel 2110   ∪ cun 3890  {csn 4567  ⟨cop 4573   ↦ cmpt 5162

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2015  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2711  ax-sep 5227  ax-nul 5234  ax-pr 5356

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2072  df-clab 2718  df-cleq 2732  df-clel 2818  df-nfc 2891  df-ne 2946  df-ral 3071  df-rex 3072  df-rab 3075  df-v 3433  df-sbc 3721  df-dif 3895  df-un 3897  df-nul 4263  df-if 4466  df-sn 4568  df-pr 4570  df-op 4574  df-opab 5142  df-mpt 5163

This theorem is referenced by:  fmptpr  7039  poimirlem3  35768  poimirlem4  35769  poimirlem16  35781  poimirlem17  35782  poimirlem19  35784  poimirlem20  35785